Properties

 Label 4-1148e2-1.1-c1e2-0-9 Degree $4$ Conductor $1317904$ Sign $1$ Analytic cond. $84.0307$ Root an. cond. $3.02767$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $2$

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Dirichlet series

 L(s)  = 1 − 2·2-s + 2·4-s − 7-s − 4·9-s − 9·11-s + 2·14-s − 4·16-s + 8·18-s + 18·22-s + 4·23-s + 2·25-s − 2·28-s + 6·29-s + 8·32-s − 8·36-s − 12·37-s − 5·43-s − 18·44-s − 8·46-s − 6·49-s − 4·50-s − 15·53-s − 12·58-s + 4·63-s − 8·64-s − 9·67-s + 24·74-s + ⋯
 L(s)  = 1 − 1.41·2-s + 4-s − 0.377·7-s − 4/3·9-s − 2.71·11-s + 0.534·14-s − 16-s + 1.88·18-s + 3.83·22-s + 0.834·23-s + 2/5·25-s − 0.377·28-s + 1.11·29-s + 1.41·32-s − 4/3·36-s − 1.97·37-s − 0.762·43-s − 2.71·44-s − 1.17·46-s − 6/7·49-s − 0.565·50-s − 2.06·53-s − 1.57·58-s + 0.503·63-s − 64-s − 1.09·67-s + 2.78·74-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$1317904$$    =    $$2^{4} \cdot 7^{2} \cdot 41^{2}$$ Sign: $1$ Analytic conductor: $$84.0307$$ Root analytic conductor: $$3.02767$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{1317904} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1317904,\ (\ :1/2, 1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + p T + p T^{2}$$
7$C_2$ $$1 + T + p T^{2}$$
41$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good3$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
5$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
13$C_2^2$ $$1 + 8 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 - 11 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + p T^{2} )$$
31$C_2^2$ $$1 - 37 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
59$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 38 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2^2$ $$1 - 75 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
83$C_2^2$ $$1 + 20 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 66 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 - 47 T^{2} + p^{2} T^{4}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

Imaginary part of the first few zeros on the critical line

−7.67335078497563088571091378899, −7.34541246957182503630415572812, −6.83498527129197968027715848030, −6.31267361837488764541791580866, −5.92040950196015795642402819102, −5.18741352189353919318546131999, −4.97798032650342694038900051396, −4.71033769091101380316359591352, −3.51887916480050099952252418650, −3.06192101833268841070385132312, −2.69224722252436758931654889310, −2.18035164361485404411932544620, −1.28053725312198844601612391556, 0, 0, 1.28053725312198844601612391556, 2.18035164361485404411932544620, 2.69224722252436758931654889310, 3.06192101833268841070385132312, 3.51887916480050099952252418650, 4.71033769091101380316359591352, 4.97798032650342694038900051396, 5.18741352189353919318546131999, 5.92040950196015795642402819102, 6.31267361837488764541791580866, 6.83498527129197968027715848030, 7.34541246957182503630415572812, 7.67335078497563088571091378899